Properties

Label 21.10.a.c
Level $21$
Weight $10$
Character orbit 21.a
Self dual yes
Analytic conductor $10.816$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.8157525594\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
Defining polynomial: \(x^{2} - x - 86\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 15 - \beta ) q^{2} -81 q^{3} + ( 58 - 30 \beta ) q^{4} + ( 564 - 70 \beta ) q^{5} + ( -1215 + 81 \beta ) q^{6} + 2401 q^{7} + ( 3540 + 4 \beta ) q^{8} + 6561 q^{9} +O(q^{10})\) \( q + ( 15 - \beta ) q^{2} -81 q^{3} + ( 58 - 30 \beta ) q^{4} + ( 564 - 70 \beta ) q^{5} + ( -1215 + 81 \beta ) q^{6} + 2401 q^{7} + ( 3540 + 4 \beta ) q^{8} + 6561 q^{9} + ( 32610 - 1614 \beta ) q^{10} + ( 36642 - 1718 \beta ) q^{11} + ( -4698 + 2430 \beta ) q^{12} + ( 70550 - 528 \beta ) q^{13} + ( 36015 - 2401 \beta ) q^{14} + ( -45684 + 5670 \beta ) q^{15} + ( 22024 + 11880 \beta ) q^{16} + ( -50892 + 14830 \beta ) q^{17} + ( 98415 - 6561 \beta ) q^{18} + ( 240872 - 2892 \beta ) q^{19} + ( 757212 - 20980 \beta ) q^{20} -194481 q^{21} + ( 1142340 - 62412 \beta ) q^{22} + ( -491106 + 95690 \beta ) q^{23} + ( -286740 - 324 \beta ) q^{24} + ( 55471 - 78960 \beta ) q^{25} + ( 1240410 - 78470 \beta ) q^{26} -531441 q^{27} + ( 139258 - 72030 \beta ) q^{28} + ( -1275462 + 142260 \beta ) q^{29} + ( -2641410 + 130734 \beta ) q^{30} + ( -2467924 - 196092 \beta ) q^{31} + ( -5580720 + 154128 \beta ) q^{32} + ( -2968002 + 139158 \beta ) q^{33} + ( -5879730 + 273342 \beta ) q^{34} + ( 1354164 - 168070 \beta ) q^{35} + ( 380538 - 196830 \beta ) q^{36} + ( -8128258 + 442296 \beta ) q^{37} + ( 4610820 - 284252 \beta ) q^{38} + ( -5714550 + 42768 \beta ) q^{39} + ( 1899960 - 245544 \beta ) q^{40} + ( 24353928 - 196590 \beta ) q^{41} + ( -2917215 + 194481 \beta ) q^{42} + ( 3994820 + 168744 \beta ) q^{43} + ( 19906536 - 1198904 \beta ) q^{44} + ( 3700404 - 459270 \beta ) q^{45} + ( -40379640 + 1926456 \beta ) q^{46} + ( 42786204 + 768164 \beta ) q^{47} + ( -1783944 - 962280 \beta ) q^{48} + 5764801 q^{49} + ( 28073265 - 1239871 \beta ) q^{50} + ( 4122252 - 1201230 \beta ) q^{51} + ( 9556700 - 2147124 \beta ) q^{52} + ( 13282662 + 3803152 \beta ) q^{53} + ( -7971615 + 531441 \beta ) q^{54} + ( 62155788 - 3533892 \beta ) q^{55} + ( 8499540 + 9604 \beta ) q^{56} + ( -19510632 + 234252 \beta ) q^{57} + ( -68211630 + 3409362 \beta ) q^{58} + ( 57600480 + 1648036 \beta ) q^{59} + ( -61334172 + 1699380 \beta ) q^{60} + ( -71410102 + 6373980 \beta ) q^{61} + ( 30632880 - 473456 \beta ) q^{62} + 15752961 q^{63} + ( -148161248 + 1810080 \beta ) q^{64} + ( 52541400 - 5236292 \beta ) q^{65} + ( -92529540 + 5055372 \beta ) q^{66} + ( 13760696 - 5935140 \beta ) q^{67} + ( -156442236 + 2386900 \beta ) q^{68} + ( 39779586 - 7750890 \beta ) q^{69} + ( 78296610 - 3875214 \beta ) q^{70} + ( 19035090 - 19134282 \beta ) q^{71} + ( 23225940 + 26244 \beta ) q^{72} + ( -1047658 - 7076172 \beta ) q^{73} + ( -274515990 + 14762698 \beta ) q^{74} + ( -4493151 + 6395760 \beta ) q^{75} + ( 43902776 - 7393896 \beta ) q^{76} + ( 87977442 - 4124918 \beta ) q^{77} + ( -100473210 + 6356070 \beta ) q^{78} + ( 217548524 + 20624004 \beta ) q^{79} + ( -274480464 + 5158640 \beta ) q^{80} + 43046721 q^{81} + ( 433132470 - 27302778 \beta ) q^{82} + ( 132144372 + 34502856 \beta ) q^{83} + ( -11279898 + 5834430 \beta ) q^{84} + ( -386847588 + 11926560 \beta ) q^{85} + ( 1705620 - 1463660 \beta ) q^{86} + ( 103312422 - 11523060 \beta ) q^{87} + ( 127341840 - 5935152 \beta ) q^{88} + ( 321336888 - 11924350 \beta ) q^{89} + ( 213954210 - 10589454 \beta ) q^{90} + ( 169390550 - 1267728 \beta ) q^{91} + ( -1018875648 + 20283200 \beta ) q^{92} + ( 199901844 + 15883452 \beta ) q^{93} + ( 376776480 - 31263744 \beta ) q^{94} + ( 205693608 - 18492128 \beta ) q^{95} + ( 452038320 - 12484368 \beta ) q^{96} + ( 172680614 + 38859852 \beta ) q^{97} + ( 86472015 - 5764801 \beta ) q^{98} + ( 240408162 - 11271798 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 30q^{2} - 162q^{3} + 116q^{4} + 1128q^{5} - 2430q^{6} + 4802q^{7} + 7080q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q + 30q^{2} - 162q^{3} + 116q^{4} + 1128q^{5} - 2430q^{6} + 4802q^{7} + 7080q^{8} + 13122q^{9} + 65220q^{10} + 73284q^{11} - 9396q^{12} + 141100q^{13} + 72030q^{14} - 91368q^{15} + 44048q^{16} - 101784q^{17} + 196830q^{18} + 481744q^{19} + 1514424q^{20} - 388962q^{21} + 2284680q^{22} - 982212q^{23} - 573480q^{24} + 110942q^{25} + 2480820q^{26} - 1062882q^{27} + 278516q^{28} - 2550924q^{29} - 5282820q^{30} - 4935848q^{31} - 11161440q^{32} - 5936004q^{33} - 11759460q^{34} + 2708328q^{35} + 761076q^{36} - 16256516q^{37} + 9221640q^{38} - 11429100q^{39} + 3799920q^{40} + 48707856q^{41} - 5834430q^{42} + 7989640q^{43} + 39813072q^{44} + 7400808q^{45} - 80759280q^{46} + 85572408q^{47} - 3567888q^{48} + 11529602q^{49} + 56146530q^{50} + 8244504q^{51} + 19113400q^{52} + 26565324q^{53} - 15943230q^{54} + 124311576q^{55} + 16999080q^{56} - 39021264q^{57} - 136423260q^{58} + 115200960q^{59} - 122668344q^{60} - 142820204q^{61} + 61265760q^{62} + 31505922q^{63} - 296322496q^{64} + 105082800q^{65} - 185059080q^{66} + 27521392q^{67} - 312884472q^{68} + 79559172q^{69} + 156593220q^{70} + 38070180q^{71} + 46451880q^{72} - 2095316q^{73} - 549031980q^{74} - 8986302q^{75} + 87805552q^{76} + 175954884q^{77} - 200946420q^{78} + 435097048q^{79} - 548960928q^{80} + 86093442q^{81} + 866264940q^{82} + 264288744q^{83} - 22559796q^{84} - 773695176q^{85} + 3411240q^{86} + 206624844q^{87} + 254683680q^{88} + 642673776q^{89} + 427908420q^{90} + 338781100q^{91} - 2037751296q^{92} + 399803688q^{93} + 753552960q^{94} + 411387216q^{95} + 904076640q^{96} + 345361228q^{97} + 172944030q^{98} + 480816324q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.78709
−8.78709
−3.57418 −81.0000 −499.225 −736.192 289.508 2401.00 3614.30 6561.00 2631.28
1.2 33.5742 −81.0000 615.225 1864.19 −2719.51 2401.00 3465.70 6561.00 62588.7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.10.a.c 2
3.b odd 2 1 63.10.a.b 2
4.b odd 2 1 336.10.a.l 2
7.b odd 2 1 147.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.a.c 2 1.a even 1 1 trivial
63.10.a.b 2 3.b odd 2 1
147.10.a.e 2 7.b odd 2 1
336.10.a.l 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 30 T_{2} - 120 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -120 - 30 T + T^{2} \)
$3$ \( ( 81 + T )^{2} \)
$5$ \( -1372404 - 1128 T + T^{2} \)
$7$ \( ( -2401 + T )^{2} \)
$11$ \( 324360384 - 73284 T + T^{2} \)
$13$ \( 4881122020 - 141100 T + T^{2} \)
$17$ \( -73285474836 + 101784 T + T^{2} \)
$19$ \( 55133856304 - 481744 T + T^{2} \)
$23$ \( -2917833651264 + 982212 T + T^{2} \)
$29$ \( -5355274808556 + 2550924 T + T^{2} \)
$31$ \( -7175316130304 + 4935848 T + T^{2} \)
$37$ \( -1422306192956 + 16256516 T + T^{2} \)
$41$ \( 579780377334684 - 48707856 T + T^{2} \)
$43$ \( 6134871382480 - 7989640 T + T^{2} \)
$47$ \( 1627083056570496 - 85572408 T + T^{2} \)
$53$ \( -4813638861804636 - 26565324 T + T^{2} \)
$59$ \( 2380787479463280 - 115200960 T + T^{2} \)
$61$ \( -8917126591287596 + 142820204 T + T^{2} \)
$67$ \( -11963574198357584 - 27521392 T + T^{2} \)
$71$ \( -125949323289847680 - 38070180 T + T^{2} \)
$73$ \( -17273814922601516 + 2095316 T + T^{2} \)
$79$ \( -99418231347666944 - 435097048 T + T^{2} \)
$83$ \( -393242104842799536 - 264288744 T + T^{2} \)
$89$ \( 54201803181262044 - 642673776 T + T^{2} \)
$97$ \( -491161799172939884 - 345361228 T + T^{2} \)
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